They all tell a similar story: They ran apps that helped people limit the time they and their children spent on iPhones. Then Apple created its own screen-time tracker. And then Apple made staying in business very, very difficult.

Over the past year, Apple has removed or restricted at least 11 of the 17 most downloaded screen-time and parental-control apps, according to an analysis by The New York Times and Sensor Tower, an app-data firm. Apple has also clamped down on a number of lesser-known apps…

Some app makers with thousands of paying customers have shut down. Most others say their futures are in jeopardy.“They yanked us out of the blue with no warning,” said Amir Moussavian, chief executive of OurPact, the top parental-control iPhone app, with more than three million downloads. In February, Apple pulled the app, which accounted for 80 percent of OurPact’s revenue, from its App Store.